Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On the invariant density of branching diffusions

Auteur(s):

Code(s) de Classification MSC:

• 60H07 Stochastic calculus of variations and the Malliavin calculus
Résumé: We consider the invariant measure for finite systems of branching diffusions with immigration. In case of an absolute continuous immigration measure, we use the properties of the underlying stochastic flow governing the motion of every particle in order to show smoothness of the invariant density of the particle process. In case of a singular immigration measure, the main tool is Malliavin calculus which allows to show that the density is $C^{\infty}$ in any point which is not an atom of the immigration measure.